The table below shows three values of x and their corresponding values of y, where \(\mathrm{y = 2f(x) - 1}\)...

1. TRANSLATE the problem information

• Given information:
  • f(x) is quadratic, so \(\mathrm{f(x) = ax^2 + bx + c}\) for some constants a, b, c
  • \(\mathrm{y = 2f(x) - 1}\), which means \(\mathrm{y = 2(ax^2 + bx + c) - 1 = 2ax^2 + 2bx + 2c - 1}\)
  • Data points: \(\mathrm{(10, 23), (12, 15), (14, 23)}\)

2. TRANSLATE each data point into an equation

• For \(\mathrm{x = 10, y = 23}\): \(\mathrm{23 = 2a(100) + 2b(10) + 2c - 1}\)
  Simplifying: \(\mathrm{24 = 200a + 20b + 2c}\), or \(\mathrm{12 = 100a + 10b + c}\)
• For \(\mathrm{x = 12, y = 15}\): \(\mathrm{15 = 2a(144) + 2b(12) + 2c - 1}\)
  Simplifying: \(\mathrm{16 = 288a + 24b + 2c}\), or \(\mathrm{8 = 144a + 12b + c}\)
• For \(\mathrm{x = 14, y = 23}\): \(\mathrm{23 = 2a(196) + 2b(14) + 2c - 1}\)
  Simplifying: \(\mathrm{24 = 392a + 28b + 2c}\), or \(\mathrm{12 = 196a + 14b + c}\)

3. INFER the solution strategy

• We now have three linear equations in three unknowns (a, b, c)
• Use elimination method to solve the system systematically

4. SIMPLIFY using elimination method

• From equation (1) - equation (2): \(\mathrm{12 - 8 = (100a + 10b + c) - (144a + 12b + c)}\)
  This gives: \(\mathrm{4 = -44a - 2b}\), so \(\mathrm{b = -22a - 2}\)
• From equation (3) - equation (1): \(\mathrm{12 - 12 = (196a + 14b + c) - (100a + 10b + c)}\)
  This gives: \(\mathrm{0 = 96a + 4b}\), so \(\mathrm{b = -24a}\)
• Setting the expressions for b equal: \(\mathrm{-22a - 2 = -24a}\)
  Solving: \(\mathrm{-2 = -2a}\), so \(\mathrm{a = 1}\)

5. SIMPLIFY to find remaining coefficients

• Since \(\mathrm{a = 1}\): \(\mathrm{b = -24(1) = -24}\)
• Substitute into equation (1): \(\mathrm{12 = 100(1) + 10(-24) + c}\)
  \(\mathrm{12 = 100 - 240 + c}\), so \(\mathrm{c = 152}\)

6. INFER the final answer

• We found \(\mathrm{f(x) = x^2 - 24x + 152}\)
• The y-intercept occurs when \(\mathrm{x = 0}\): \(\mathrm{f(0) = 0^2 - 24(0) + 152 = 152}\)

Answer: C) 152

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly set up the initial equations from the data points, often forgetting to properly substitute into \(\mathrm{y = 2f(x) - 1}\) or making sign errors in the algebraic manipulation.

For example, they might write \(\mathrm{23 = 2a(10)^2 + 2b(10) + 2c}\) instead of \(\mathrm{23 = 2a(10)^2 + 2b(10) + 2c - 1}\), missing the "-1" term. This leads to a completely different system of equations and wrong coefficients.

This may lead them to select Choice A (150) or get confused and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors while solving the system of equations, particularly in the elimination steps or when substituting back to find the coefficients.

Common mistakes include sign errors (like getting \(\mathrm{a = -1}\) instead of \(\mathrm{a = 1}\)) or calculation errors when computing c. These small errors compound to give wrong final answers.

This may lead them to select Choice B (151) or Choice D (153).

The Bottom Line:

This problem challenges students to work backwards from transformed data to reconstruct the original quadratic function. Success requires careful translation of the transformation relationship and systematic algebraic manipulation without computational errors.